At the climax of Samurai Champloo, Jin's path to victory lies in the willingness to make an emotional decision to sacrifice himself. Mugen, meanwhile, actually has to think about his attack. Both of them have to do exactly the opposite of their personalities.
One of the biggest questions about the predecessor, Cowboy Bebop, is whether Spike really died in the end. His expy, Mugen has actually come back from the dead more than once, though in his case he refused to submit to the Grim Reaper of his afterlife.
In the Brick Joke at the very end where Fuu reveals the coin had been heads the whole time, didn't both Mugen and Jin have a nice long look at the coin and would have seen this? - It landed on her head and she took it off before the other two could see it, so she probably flipped it.
How does Mugen survive being killed so many times? Answer: Mushrooms!
You know, the episode with the slave traders is kind of unsettling (that is, aside from the obvious human trafficking plot). Only one criminal was apprehended, hinting that the others might as well continue their illicit trade. And the barrels contain the hostages right? Several fell into the water...
How did Mugen learn ki blasts so quickly? He already knew how to pull it off to an extent, he cuts a skewer in half using only a wind blast in episode 2.
